More Doreyan sequences.

[Antti Karttunen]

Antti Karttunen wrote:
> 
> "N. J. A. Sloane" wrote:
> >
> 
> > >From laurentdorey at wanadoo.fr  Thu Feb 20 21:10:44 2003
> > >I can give you an example of the kind of things I am looking for : I did this one last week
> 
> > >-01-02-03-04-05-06-07-08-09-10-01-03-07-05-01-03-10-01-03-07-05-01-03-10-01-04-09-09-09-09-08-
> > 07-05-06-03-07-02-05-01-05-01-03-07-05-01-04-03-03-07-02-07-09-09-09-08-05-01-03-10-01-04-09-08-07-02-05-06- and so on...
> > >   > > *******************************+02+04+08+06+02+07+01+02+04+08+06+02+07+01+03+05+10+10+10+09+09+
> > 08+01+07+04+05+03+06+04+06+02+04+08+06+03+09+10+04+05+05+02+10+10+09+07+06+02+07+01+03+05+09+09+05+03+01+07+ and so on...


Me, concerning A080867 & A080868:

> Maybe one could take alternatively the summand from the first and
> the last possible positions, or use other "heuristic" to break
> the periodicity.

OK, here I implemented the last idea, taking alternatively
the first (i.e. simply 2*a(n-1)+1 modulo 10)
or the last occurrence:

A0xxxxx := proc(n) option remember; local i,u;
  if(n < 12) then RETURN(n mod 10);
  else
   u := A0xxxxx(n-1);
   if(0 = (n mod 2)) then RETURN((2*u+1) mod 10);
   else
     for i from n-2 by -1 to 0
      do
         if(A0xxxxx(i) = u) then RETURN((A0xxxxx(i+1)+u) mod 10); fi;
      od;
   fi
  fi;
  RETURN(-1); # Jamais!
end;


Anybody wants to check what is its period?
Note that this now matches with Laurent's original
sequence above up to the 18th term (mod 10).


[seq(A0xxxxx(n),n=0..1000)];
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 3, 7, 5, 1, 3, 0, 1, 4, 9, 9, 9,

    8, 7, 2, 5, 6, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1, 6, 3, 3, 7,

    3, 7, 0, 1, 7, 5, 6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7, 9, 9, 8, 7,

    6, 3, 0, 1, 4, 9, 7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1, 4, 9, 6, 3,

    3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1, 6, 3, 3, 7, 3, 7, 0, 1, 7, 5,

    6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 4, 9,

    7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1, 4, 9, 6, 3, 3, 7, 9, 9, 8, 7,

    6, 3, 0, 1, 5, 1, 6, 3, 3, 7, 3, 7, 0, 1, 7, 5, 6, 3, 0, 1, 8, 7,

    2, 5, 1, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 4, 9, 7, 5, 6, 3, 3, 7,

    2, 5, 1, 3, 0, 1, 4, 9, 6, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1,

    6, 3, 3, 7, 3, 7, 0, 1, 7, 5, 6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7,

    9, 9, 8, 7, 6, 3, 0, 1, 4, 9, 7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1,

    4, 9, 6, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1, 6, 3, 3, 7, 3, 7,

    0, 1, 7, 5, 6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7, 9, 9, 8, 7, 6, 3,

    0, 1, 4, 9, 7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1, 4, 9, 6, 3, 3, 7,

    9, 9, 8, 7, 6, 3, 0, 1, 5, 1, 6, 3, 3, 7, 3, 7, 0, 1, 7, 5, 6, 3,

    0, 1, 8, 7, 2, 5, 1, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 4, 9, 7, 5,

    6, 3, 3, 7, 2, 5, 1, 3, 0, 1, 4, 9, 6, 3, 3, 7, 9, 9, 8, 7, 6, 3,

    0, 1, 5, 1, 6, 3, 3, 7, 3, 7, 0, 1, 7, 5, 6, 3, 0, 1, 8, 7, 2, 5,

    1, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 4, 9, 7, 5, 6, 3, 3, 7, 2, 5,

    1, 3, 0, 1, 4, 9, 6, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1, 6, 3,

    3, 7, 3, 7, 0, 1, 7, 5, 6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7, 9, 9,

    8, 7, 6, 3, 0, 1, 4, 9, 7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1, 4, 9,

    6, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1, 6, 3, 3, 7, 3, 7, 0, 1,

    7, 5, 6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1,

    4, 9, 7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1, 4, 9, 6, 3, 3, 7, 9, 9,

    8, 7, 6, 3, 0, 1, 5, 1, 6, 3, 3, 7, 3, 7, 0, 1, 7, 5, 6, 3, 0, 1,

    8, 7, 2, 5, 1, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 4, 9, 7, 5, 6, 3,

    3, 7, 2, 5, 1, 3, 0, 1, 4, 9, 6, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1,

    5, 1, 6, 3, 3, 7, 3, 7, 0, 1, 7, 5, 6, 3, 0, 1, 8, 7, 2, 5, 1, 3,

    3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 4, 9, 7, 5, 6, 3, 3, 7, 2, 5, 1, 3,

    0, 1, 4, 9, 6, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1, 6, 3, 3, 7,

    3, 7, 0, 1, 7, 5, 6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7, 9, 9, 8, 7,

    6, 3, 0, 1, 4, 9, 7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1, 4, 9, 6, 3,

    3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1, 6, 3, 3, 7, 3, 7, 0, 1, 7, 5,

    6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 4, 9,

    7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1, 4, 9, 6, 3, 3, 7, 9, 9, 8, 7,

    6, 3, 0, 1, 5, 1, 6, 3, 3, 7, 3, 7, 0, 1, 7, 5, 6, 3, 0, 1, 8, 7,

    2, 5, 1, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 4, 9, 7, 5, 6, 3, 3, 7,

    2, 5, 1, 3, 0, 1, 4, 9, 6, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1,

    6, 3, 3, 7, 3, 7, 0, 1, 7, 5, 6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7,

    9, 9, 8, 7, 6, 3, 0, 1, 4, 9, 7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1,

    4, 9, 6, 3, 3, 7, 9, 9, 8, 7, 6, 3, 0, 1, 5, 1, 6, 3, 3, 7, 3, 7,

    0, 1, 7, 5, 6, 3, 0, 1, 8, 7, 2, 5, 1, 3, 3, 7, 9, 9, 8, 7, 6, 3,

    0, 1, 4, 9, 7, 5, 6, 3, 3, 7, 2, 5, 1, 3, 0, 1, 4, 9, 6, 3, 3, 7,

    9, 9, 8, 7, 6, 3, 0, 1, 5, 1]

> mod10 := n -> n mod 10;
> map(mod10,DIFF([seq(A0xxxxx(n),n=0..1000)]));

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 6, 2, 7, 1, 3, 5, 0, 0, 9,

    9, 5, 3, 1, 7, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5, 7, 0, 4, 6,

    4, 3, 1, 6, 8, 1, 7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2, 0, 9, 9, 9,

    7, 7, 1, 3, 5, 8, 8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3, 5, 7, 7, 0,

    4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5, 7, 0, 4, 6, 4, 3, 1, 6, 8, 1,

    7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 3, 5, 8,

    8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3, 5, 7, 7, 0, 4, 2, 0, 9, 9, 9,

    7, 7, 1, 4, 6, 5, 7, 0, 4, 6, 4, 3, 1, 6, 8, 1, 7, 7, 1, 7, 9, 5,

    3, 6, 2, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 3, 5, 8, 8, 1, 7, 0, 4, 5,

    3, 6, 2, 7, 1, 3, 5, 7, 7, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5,

    7, 0, 4, 6, 4, 3, 1, 6, 8, 1, 7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2,

    0, 9, 9, 9, 7, 7, 1, 3, 5, 8, 8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3,

    5, 7, 7, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5, 7, 0, 4, 6, 4, 3,

    1, 6, 8, 1, 7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2, 0, 9, 9, 9, 7, 7,

    1, 3, 5, 8, 8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3, 5, 7, 7, 0, 4, 2,

    0, 9, 9, 9, 7, 7, 1, 4, 6, 5, 7, 0, 4, 6, 4, 3, 1, 6, 8, 1, 7, 7,

    1, 7, 9, 5, 3, 6, 2, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 3, 5, 8, 8, 1,

    7, 0, 4, 5, 3, 6, 2, 7, 1, 3, 5, 7, 7, 0, 4, 2, 0, 9, 9, 9, 7, 7,

    1, 4, 6, 5, 7, 0, 4, 6, 4, 3, 1, 6, 8, 1, 7, 7, 1, 7, 9, 5, 3, 6,

    2, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 3, 5, 8, 8, 1, 7, 0, 4, 5, 3, 6,

    2, 7, 1, 3, 5, 7, 7, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5, 7, 0,

    4, 6, 4, 3, 1, 6, 8, 1, 7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2, 0, 9,

    9, 9, 7, 7, 1, 3, 5, 8, 8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3, 5, 7,

    7, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5, 7, 0, 4, 6, 4, 3, 1, 6,

    8, 1, 7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 3,

    5, 8, 8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3, 5, 7, 7, 0, 4, 2, 0, 9,

    9, 9, 7, 7, 1, 4, 6, 5, 7, 0, 4, 6, 4, 3, 1, 6, 8, 1, 7, 7, 1, 7,

    9, 5, 3, 6, 2, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 3, 5, 8, 8, 1, 7, 0,

    4, 5, 3, 6, 2, 7, 1, 3, 5, 7, 7, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 4,

    6, 5, 7, 0, 4, 6, 4, 3, 1, 6, 8, 1, 7, 7, 1, 7, 9, 5, 3, 6, 2, 0,

    4, 2, 0, 9, 9, 9, 7, 7, 1, 3, 5, 8, 8, 1, 7, 0, 4, 5, 3, 6, 2, 7,

    1, 3, 5, 7, 7, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5, 7, 0, 4, 6,

    4, 3, 1, 6, 8, 1, 7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2, 0, 9, 9, 9,

    7, 7, 1, 3, 5, 8, 8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3, 5, 7, 7, 0,

    4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5, 7, 0, 4, 6, 4, 3, 1, 6, 8, 1,

    7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 3, 5, 8,

    8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3, 5, 7, 7, 0, 4, 2, 0, 9, 9, 9,

    7, 7, 1, 4, 6, 5, 7, 0, 4, 6, 4, 3, 1, 6, 8, 1, 7, 7, 1, 7, 9, 5,

    3, 6, 2, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 3, 5, 8, 8, 1, 7, 0, 4, 5,

    3, 6, 2, 7, 1, 3, 5, 7, 7, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5,

    7, 0, 4, 6, 4, 3, 1, 6, 8, 1, 7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2,

    0, 9, 9, 9, 7, 7, 1, 3, 5, 8, 8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3,

    5, 7, 7, 0, 4, 2, 0, 9, 9, 9, 7, 7, 1, 4, 6, 5, 7, 0, 4, 6, 4, 3,

    1, 6, 8, 1, 7, 7, 1, 7, 9, 5, 3, 6, 2, 0, 4, 2, 0, 9, 9, 9, 7, 7,

    1, 3, 5, 8, 8, 1, 7, 0, 4, 5, 3, 6, 2, 7, 1, 3, 5, 7, 7, 0, 4, 2,

    0, 9, 9, 9, 7, 7, 1, 4, 6]


Terveisin,
 Salut,

Antti Karttunen